Let's take a detailed look at the given question and each of its parts. We need to use the coefficients [tex]\( K_i \)[/tex] from the binomial expansion of [tex]\( (x+y)^4 \)[/tex].
1. Finding the sum [tex]\( \sum_{i=1}^5 K_i \)[/tex]:
The expression [tex]\((x + y)^4\)[/tex] can be expanded using the Binomial Theorem. The expanded form is:
[tex]\[
(x + y)^4 = \sum_{i=0}^4 \binom{4}{i} x^{4-i} y^i
\][/tex]
This gives us the coefficients [tex]\( K_1, K_2, K_3, K_4, K_5 \)[/tex] as [tex]\(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4} \)[/tex]:
[tex]\[
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
\][/tex]
The coefficients are:
[tex]\[
K_1 = 1, \, K_2 = 4, \, K_3 = 6, \, K_4 = 4, \, K_5 = 1
\][/tex]
Sum of all coefficients [tex]\( \sum_{i=1}^5 K_i \)[/tex]:
[tex]\[
K_1 + K_2 + K_3 + K_4 + K_5 = 1 + 4 + 6 + 4 + 1 = 16
\][/tex]
2. Finding [tex]\( \sum_{i=2}^4 K_i \)[/tex]:
We need to sum the coefficients from [tex]\( K_2 \)[/tex] to [tex]\( K_4 \)[/tex]:
[tex]\[
K_2 + K_3 + K_4 = 4 + 6 + 4 = 14
\][/tex]
3. Finding [tex]\( \sum_{i=1}^4 K_i \)[/tex]:
We need to sum the coefficients from [tex]\( K_1 \)[/tex] to [tex]\( K_4 \)[/tex]:
[tex]\[
K_1 + K_2 + K_3 + K_4 = 1 + 4 + 6 + 4 = 15
\][/tex]
4. Finding [tex]\( \sum_{i=3}^5 K_i \)[/tex]:
We need to sum the coefficients from [tex]\( K_3 \)[/tex] to [tex]\( K_5 \)[/tex]:
[tex]\[
K_3 + K_4 + K_5 = 6 + 4 + 1 = 11
\][/tex]
Now, let's use these sums to calculate the values for [tex]\( q \)[/tex].
(i) Calculating [tex]\( 4 \sum_{i=1}^5 K_1 \)[/tex]:
We have already calculated [tex]\( \sum_{i=1}^5 K_i = 16 \)[/tex]:
[tex]\[
4 \sum_{i=1}^5 K_1 = 4 \times 16 = 64
\][/tex]
(iii) Calculating [tex]\( 2 \sum_{i=2}^4 K_i \)[/tex]:
We have already calculated [tex]\( \sum_{i=2}^4 K_i = 14 \)[/tex]:
[tex]\[
2 \sum_{i=2}^4 K_i = 2 \times 14 = 28
\][/tex]
(iii) Calculating [tex]\( 3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i \)[/tex]:
We have already calculated [tex]\( \sum_{i=1}^4 K_i = 15 \)[/tex] and [tex]\( \sum_{i=3}^5 K_i = 11 \)[/tex]:
[tex]\[
3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i = 3 \times 15 - 2 \times 11 = 45 - 22 = 23
\][/tex]
Thus, the values for [tex]\( q \)[/tex] are:
1. [tex]\( 4 \sum_{i=1}^5 K_1 = 64 \)[/tex]
2. [tex]\( 2 \sum_{i=2}^4 K_i = 28 \)[/tex]
3. [tex]\( 3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i = 23 \)[/tex]
So, the answers are:
[tex]\((i) \, q_1 = 64\)[/tex]
[tex]\((iii) \, q_3 = 28\)[/tex]
[tex]\((iii \text{ alt}) \, q_3 = 23\)[/tex]
